Multi-link cubes are an incredibly versatile manipulative for mathematics class. You may see these 2 cm interlocking cubes referred to by several different names including: linking cubes, multi-link cubes, Snap cubes, Cube-a-Links and Hex-a-Link. It's a manipulative that I've seen used at nearly every grade level from Primary to 12. Recently, John Rowe (@MrJohnRowe) stated a conversation on Twitter about math manipulatives. The conversation prompted me to reflect on how I've used manipulatives, and especially multi-link cubes, as part of instruction and inquiry. I thought it would be nice to list a few of my favourite examples here:

Speedy Squares from Mary Bourassa (@MaryBourassa) Speedy Squares is an activity that asks students to predict how long it would take them to build a 26 x 26 square out of linking cubes. Students start by building smaller squares and recording their times. They can then use this time to extrapolate an answer. Students could use quadratic regression to make a more accurate prediction. Jon Orr (@MrOrr_geek) also blogged about this activity and how he introduced it to his class. ​NS Outcomes: Math Extended 11 RF02, PCAL 11 RF04 and Math 12 RF01

​Hotel Snap from Fawn Nguyen (@fawnpnguyen)In this activity, students design a hotel using multi-link cubes. Students are given a number or restrictions on building as well as costs to minimize and profits to maximize. I blogged about this activity, that I called Hotel Cubed.NS Outcomes: Math 9 G01, Math 10 FM01 and Math at Work 10 N01

Orthographic Projections from Jocelle Skov (@mrs_skov)Each student creates a 3D object using multi-link cubes. Next they draw the top, front and side views of their object. Once every student has finished the three views of their object, they trade drawings with another student. That student then tries to build the 3D object in the drawing. They check their work with the original object on the teachers desk. NS Outcomes: Math 8 G01, Math at Work 11 G03

Photo from Jocelle Skov

Photo from Jocelle Skov

3D Linear Relations inspired by Alicia Potvin (@AliciaPotvin1)Each group of students builds three terms of a linear pattern using multi-link cubes. Ask students to use one colour for the part of the pattern that stays the same and another colour for the part of the pattern that changes. Groups then rotate through the room and for each pattern, record a table of values, a graph and the equation. You could also ask students to determine how many cubes would be in the 43rd term as suggested at http://www.visualpatterns.org/.NS Outcomes: Math 9 PR01, Math 10 RF04

t(n) = 2n + 1

t(n) = 3n + 2

The Condo Challenge from Marilyn BurnsStudents are asked how many cubes are required to build a "Condo" that is 6-cubes high. Then they predict how many cubes would be required for a condo 12-cubes high. Later they create a mathematical model to predict the number of cubes for a condo of any height. Thanks to Halcyon Foster for suggesting this task! Although she doesn't mention linking cubes, Jo Boaler's Growing Shapes task (from her week of inspirational math) is a similar problem.​NS Outcomes: Math 9 PR01, Math 10 RF04

Photo from Halcyon Foster

Mean, Median, Mode and Range with Linking Cubes from Jana Barnard and Cathy TalleyAsk each student to reach into a large box of linking cubes to grab as many as they can with one hand. Students then build a tower with their linking cubes. As a class, students organize their towers in order from shortest to tallest. To get the class range, subtract the height of the shortest tower from the tallest tower. Is there a height that occurs more often than any other? That is the mode. To get the median, find the tower in the middle of the row (if an even amount of towers, average the two middle towers). To get the mean, even out all the towers until they are the same height saving any "left over". Suppose you had 12 towers, each with a height of 10 and 5 remainder cubes. This would give a mean of 10 and 5/12 cubes.NS Outcomes: Grade 7 SP01

Spinners from Shaun Mitchell and MikeWiernicki (h/t Jen Carter)Students, in small groups, design a spinning top made of multi-link cubes. The goal is to design a top that spins the longest. Once the group settles on their design they collect some data. They spin the top and record the time it spins in seconds to the nearest hundredth (or tenth). They do this three or four times and then average the time (hence they have to add three or four decimal numbers and then divide that decimal by 3 or 4). They could also model their decimal numbers using decimal squares. NS Outcomes: Grade 6 N08, Grade 6 SP02

Image from Jen Carter

A few links to some documents that provide some additional suggestions for using linking cubes:

I was recently invited by a class to work with them on collecting and analysing data. After brainstorming some ideas with the classroom teacher, we settled on collecting data from pull back cars. I checked out Fawn Nguyen's Vroom Vroom lesson and Simon Job's Car Racing lesson to get some ideas on how to organize this lesson. We started the lesson by sharing the first half of Simon's video of cars racing across the floor. We had the students do some notice and wonder about the action taking place in the video and then introduced activity.

We showed students the recording sheet that we would be using and how we would be taking measurements (A link to the record sheet Google Doc is here). Then we brainstormed some ways to make sure that we all collected good data and avoided errors: we would all use the same units (centimeters), all measure our distances the same way (from the front bumper), not use data if the car bumped into a wall or a desk, etc. We split up into racing teams of three students each. Each group got a measuring tape, a pull back car and a recording sheet on a clip board.

The classroom teacher and I circulated the room (and a bit of the hallway) to help students and answer questions. After students finished collecting their data and plotting their values we came back together as a class. We asked several groups to plot their data on the whiteboard at the front of the room. We then had a discussion about general trends as well as why each car had a slightly different graph. Cars might have different wind up springs, different tire grip, dusty floors, aerodynamics, etc.

We finished the class with a bit of excitement... the 150 Challenge. Each team had to use the data for their car to predict how much they would need to pull back to make the car travel as close to 150 cm as possible. Teams huddled to interpret their data and select a pull back distance. Each team brought their car to the front of the class to give it their best shot. There was lots of cheering and excitement as some teams got very close. The winning distance was only 2.5 cm! Much closer than I had expected. This activity could be easily extended for higher grade levels by incorporating linear relationships, linear equations and linear regression.

Nova Scotia Mathematics Curriculum OutcomesMathematics 6 SP01 - Students will be expected to create, label, and interpret line graphs to draw conclusions. Mathematics 6 SP02 - Students will be expected to select, justify, and use appropriate methods of collecting data, including questionnaires, experiments, databases, and electronic media.Mathematics 6 SP03 - Students will be expected to graph collected data and analyze the graph to solve problems. Mathematics 9 PR02 - Students will be expected to graph a linear relation, analyze the graph, and interpolate or extrapolate to solve problems.Mathematics 10 RF07 - Determine the equation of a linear relation, given: a graph, a point and the slope, two points and a point and the equation of a parallel or perpendicular line to solve problems. (including RF07.06 Determine the equation of the line of best fit from a scatterplot using technology and determine the correlation)Mathematics Extended 11 S01 - Analyze, interpret, and draw conclusions from one-variable data using numerical and graphical summaries.